3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. √(x^4 + x^3)

Verified step by step guidance

1. Rewrite the function to better understand its growth rate. The function is \(\sqrt{x^4 + x^3}\). Since the square root is the same as raising to the power \(\frac{1}{2}\), rewrite it as \(\left(x^4 + x^3\right)^{\frac{1}{2}}\).
2. Factor out the highest power of \(x\) inside the parentheses to simplify the expression. The highest power inside is \(x^4\), so factor it out: \(\left(x^4\left(1 + \frac{1}{x}\right)\right)^{\frac{1}{2}}\).
3. Use the property of exponents to separate the factors: \(\left(x^4\right)^{\frac{1}{2}} \cdot \left(1 + \frac{1}{x}\right)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}} \cdot \left(1 + \frac{1}{x}\right)^{\frac{1}{2}} = x^2 \cdot \left(1 + \frac{1}{x}\right)^{\frac{1}{2}}\).
4. Analyze the behavior of \(\left(1 + \frac{1}{x}\right)^{\frac{1}{2}}\) as \(x \to \infty\). Since \(\frac{1}{x} \to 0\), this term approaches \(1\).
5. Conclude that the function behaves like \(x^2\) times a term approaching \(1\), so its growth rate is asymptotically similar to \(x^2\). Therefore, it grows at the same rate as \(x^2\) as \(x \to \infty\).

Key Concepts

Asymptotic Growth Rates

Asymptotic growth rates describe how functions behave as the input variable approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate by analyzing dominant terms and their exponents.

Dominant Term in Polynomials

In polynomials, the term with the highest power of x dominates the function's behavior for large x. For example, in x^4 + x^3, the x^4 term grows faster and dictates the overall growth rate as x approaches infinity.

Simplifying Expressions with Radicals

Simplifying expressions involving square roots, such as √(x^4 + x^3), often involves factoring out the highest power inside the root to identify dominant behavior. This helps compare the function's growth rate to simpler functions like x².